Dynamic yaw steering method for spacecrafts

ABSTRACT

A method for yaw steering a spacecraft including performing yaw steering of the spacecraft to have a yaw angle (ψ) for all sun elevation angles (β). The method further including smoothing a yaw steering motion for orbital position parameters (η) where high rotational rates {dot over (ψ)} and high rotational accelerations {umlaut over (ψ)} would occur. The instant abstract is neither intended to define the invention disclosed in this specification nor intended to limit the scope of the invention in any way.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims priority under 35 U.S.C. § 119 ofEuropean Patent Application No. EP 03 024 205.1, filed on Oct. 21, 2003,the disclosure of which is expressly incorporated by reference herein inits entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method for yaw steering of a spacecraft, i.e.for satellites, space stations, and the like. Moreover the presentinvention relates to the basic idea that temporal characteristics of theacceleration about a yaw axis are continuous for all sun elevationangles β, which is defined as the angle of the sun above an orbitalplane of the spacecraft as shown in FIG. 1.

2. Discussion of Background Information

State of the art methods only provide a yaw steering for high sunelevation angles β and do not provide any yaw steering for low or zeroelevation angles at all or they show a discontinuous temporal behaviorof the acceleration about the yaw axis. Such state of the art methodsare disclosed in Barker L., Stoen J.: “Sirius satellite design: Thechallenges of the Tundra orbit in commercial spacecraft design”,Guidance and Control 2001, Proceedings of the annual AAS Rocky Mountainconference, 31 Jan. 2001, p. 575–596.

SUMMARY OF THE INVENTION

In particular, a first aspect of the invention refers to a method foryaw steering of a spacecraft, including performing yaw steering of thespacecraft yaw angle ψ for all sun elevation angles β and smoothing theyaw steering motion for values of the orbital position parameter wherehigh rotational rates {dot over (ψ)} and/or rotational accelerations{umlaut over (ψ)} would occur.

A second aspect of the invention refers to a method for yaw steering ofa spacecraft, in that yaw steering of the spacecraft yaw angle ψ iseffected for all sun elevation angles β and that a yaw steering guidancelaw is applied which is designed such that the steering motion about theyaw axis is smooth for all sun elevation angles β, whereby at least partof the guidance law comprises a smoothing function (f) being a functionof an orbital position parameter of the spacecraft, the smoothingfunction smoothing the yaw steering motion for values of the orbitalposition parameter where high rotational rates {dot over (ψ)} and/orrotational accelerations {umlaut over (ψ)} would occur withoutapplication of the smoothing function.

Preferably at least part of the yaw steering guidance law is based on asmoothing function being a function of an orbital position parameter ofthe spacecraft, where the orbital position can for example be expressedas an angle η, as shown in FIG. 1.

It can be provided that a first part of the attitude guidance law isapplied for all sun elevation angles bigger or equal to a defined sunelevation angle β₀, this first part of the guidance law beingindependent of smoothing function, and that a second part of theattitude guidance law is applied for all sun elevation angles smallerthan a defined sun elevation angle. This second part of the guidance lawis based on smoothing function. So two parts are provided for theguidance law which can be optimized separately and designed specificallyfor the respective ranges of incident angles. It can also be providedthat both parts of the guidance law are of the same form and/orstructure and/or contain the same functions, in particular the smoothingfunction, but the smoothing function being a factor equal to one or anadditive component being equal to zero in the first part of the guidancelaw.

In particular, the first part of the guidance law can comprise astandard yaw steering law ψ=atan 2(tan β,sin η) being a function of thesun elevation angle β and the orbital position parameter η, and thesecond part of the guidance law can comprise a modified yaw steering lawψ=atan 2(tan β_(d),sin η) being a function of the orbital positionparameter η and a sun elevation angle parameter β_(d) being a functionof the smoothing function and the sun elevation angle β. The defined sunelevation angle β₀ can in particular be chosen as the angle β₀ for whichthe maximum limits of the rotational rates {dot over (ψ)} and/orrotational accelerations {umlaut over (ψ)} are reached for thespacecraft when the standard yaw steering law ψ=atan 2(tan β,sin η) isapplied.

Preferably, the yaw steering motion is smoothed for orbit anglesη=k·180° (k=0, 1, 2 . . . ). It is especially for those angles that highrotational rates {dot over (ψ)} and rotational accelerations {umlautover (ψ)} can occur, so respective smoothing of the yaw steering motionis effected in order to avoid such high rotational rates {dot over (ψ)}and rotational accelerations {umlaut over (ψ)}.

It can be further provided, that the smoothing function comprises adesign parameter which can be adapted in order to optimize the smoothingeffect of smoothing function. This design parameter can be adapted oncebefore launching the guidance law or even dynamically for dynamicoptimization.

Furthermore, a smooth transient between a positive yaw steering shapefor β>0 and a negative yaw steering shape for β<0 is preferablyperformed at a sun elevation angle β≅0 and at an orbit angle η=90°. Sucha smooth transient for an angle β≅0 is desired in order to maintain asmooth yaw steering motion. The orbit angle η=90° was found to be a veryadvantageous point for such a transient.

It can be further provided that the attitude control law is designedsuch that a first panel of the satellite body structure is alwaysdirected away from the sun, or at least directed such that the incidentsun angle on that panel is very small, preferably smaller than 5°, inparticular smaller than 2°. In particular, the plane defined by thefirst panel can be oriented parallel to the yaw axis of the satellite.So for example, this panel can be the panel shown in FIG. 1 to which thepositive x-axis forms the normal.

It can be further provided that the attitude control law is designedsuch that a second and third panel of the satellite body structurealways have an incident sun angle smaller than a defined angle,preferably smaller than 5°, in particular smaller than 2°. These panelscan in particular be the panels being more or less perpendicular to they-axis as shown in FIG. 1.

One preferred smoothing function f can be chosen as:

$f = \frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}$which fulfils the above-mentioned requirements of a smoothing function.In particular, the smoothing function can be adapted by changing thedesign parameter d such that the function has very sharp maxima in avery limited region, which leads to a high smoothing effect in thatparticular region, and that the function is close to zero in a verybroad region which leads to a neglectable smoothing effect in thatregion. But the invention is not limited to this particular smoothingfunction. Other functions fulfilling these requirements can be chosenwithin the scope of the invention.

In particular, the guidance law can based on the following functions:

for|β|≧β₀:ƒ=0  (1)

and for|β|<β₀:

${(2)\mspace{50mu} f} = \frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}$β_(d)=β+ƒ·(β₀·δ−β), δ=±1  (3)ψ=atan 2(tan β_(d),sin η)  (4)

The expression of atan2 (x, y) (or arctan (x, y) which is used in theexample below in the identical sense) refers to a technically well knownfunction returning values of an inverse tangent tan⁻¹ (x/y) in theinterval of [−π, π], in contrast to atan (x/y) which returns values onlyin the interval of [−π/2, π/2].

For β=β₀ the transient between the first part of the guidance law(Kinematic Yaw Steering) and the second part of the guidance law(Dynamic Yaw Steering) is obtained in a continuous way.

By varying the design parameter d as mentioned above, it can inparticular be achieved that for |β|<β₀ and in a certain range around thecritical orbit angle η=0°, 180°, 360°, . . . , β_(d)→β₀ results. Thus,the yaw steering motion is smoothed to the behavior like for thecritical Sun elevation angle β₀, which is the “lower” limit of theKinematic Yaw Steering profile, being per definition in-line with thespacecraft actuation system capabilities.

For |β|<β₀ and in the dynamically non-critical rest period along theorbit angle, the yaw steering motion is performed as for β_(d)→β, whichis close to the standard kinematic solution.

In one aspect of the invention a method for yaw steering a spacecraftincluding performing yaw steering of the spacecraft to have a yaw angle(ψ) for all sun elevation angles (β). The method further includingsmoothing a yaw steering motion for orbital position parameters (η)where high rotational rates {dot over (ψ)} and high rotationalaccelerations {umlaut over (ψ)} would occur.

In a further aspect of the invention, high rotational rates {dot over(ψ)} can be rotational rates exceeding a predetermined rotational rateand high rotational accelerations {umlaut over (ψ)} can be rotationalaccelerations exceeding a predetermined rotational acceleration.Moreover, the yaw steering motion can be smoothed for orbit anglesη=k·180° (where k=0, 1, 2, . . . ). Additionally, the smoothing function(f) can include a design parameter (d) which is configured to optimize asmoothing effect of the smoothing function (f). Furthermore, a smoothtransient between a positive yaw steering shape for β>0 and a negativeyaw steering shape for β<0 can be performed at a sun elevation angle β≅0and at an orbit angle η=90°. Moreover, the yaw steering can be performedsuch that a first panel of a satellite body structure is always directedaway from the sun. Additionally, a plane defined by the first panel canbe oriented parallel to a yaw axis of the satellite. Furthermore, theyaw steering can be performed such that a second and a third panel of asatellite body structure always have an incident sun angle smaller thana predetermined angle. Moreover, the smoothing function (f) can bedefined as:

$f = {\frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}.}$Additionally, the guidance law can be based on the following functions:for |β|≦β₀:ƒ=0 and for |β|<β₀:  (1)

${(2)\mspace{50mu} f} = \frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}$β_(d)=β+ƒ·(β₀·δ−β), δ=±1  (3)ψ=atan 2(tan β_(d),sin η).  (4)

Yet another aspect of the invention includes a method for yaw steering aspacecraft including steering the spacecraft to have a yaw angle (ψ) forall sun elevation angles (β) in accordance with a yaw steering guidancelaw that provides a yaw steering motion about a yaw axis which is smoothfor all sun elevation angles (β). Moreover, at least part of theguidance law comprises a smoothing function (f), which is a function ofan orbital position parameter (η) of the spacecraft, the smoothingfunction (f) smoothing the yaw steering motion for values of the orbitalposition parameter (η) where high rotational rates {dot over (ψ)} orhigh rotational accelerations {umlaut over (ψ)} would occur without thesmoothing function (f).

In a further aspect of the invention high rotational rates {dot over(ψ)} can be rotational rates exceeding a predetermined rotational rateand high rotational accelerations {umlaut over (ψ)} can be rotationalaccelerations exceeding a predetermined rotational acceleration.Moreover, a first part of the guidance law can be applied for all sunelevation angles (β) larger or equal to a defined sun elevation angle(β₀), in which the first part of the guidance law is independent of thesmoothing function (f), and a second part of the guidance law is appliedfor all sun elevation angles (β) smaller than a defined sun elevationangle (β₀), in which the second part of the guidance law is based onsmoothing function (f). Furthermore, a first part of the guidance lawcan include a standard yaw steering law ψ=atan 2(tan β,sin η), which isa function of the sun elevation angle (β) and the orbital positionparameter (η), and a second part of the guidance law comprises amodified yaw steering law ψ=atan 2(tan β_(d),sin η) which is a functionof the orbital position parameter (η) and a sun elevation angleparameter (β_(d)) and is a function of the smoothing function (f) andthe sun elevation angle (β). Moreover, a spacecraft can have a yawsteering system performing the yaw steering method noted above.

A further aspect of the invention includes a method for yaw steering aspacecraft including performing yaw steering of the spacecraft to have ayaw angle for all sun elevation angles. The method further includingreducing rotational rates or rotational accelerations below apredetermined value by smoothing a yaw steering motion at predeterminedorbital position parameters.

In a further aspect of the invention the method includes steering thespacecraft with a yaw steering guidance law which provides a yawsteering motion about a yaw axis. Moreover at least part of the guidancelaw comprises a smoothing function that is a function of an orbitalposition parameter of the spacecraft, the smoothing function smoothingthe yaw steering motion. Additionally, the method can include a firstpart of the guidance law is applied for all sun elevation angles largeror equal to a defined sun elevation angle, in which the first part ofthe guidance law is independent of the smoothing function, and a secondpart of the guidance law is applied for all sun elevation angles smallerthan a defined sun elevation angle, in which the second part of theguidance law is based on smoothing function. Moreover, a spacecraft canuse the yaw steering method noted above.

The following detailed description and the corresponding figures show aspecific embodiment of the invention concerning a yaw steering methodfor satellites.

Other exemplary embodiments and advantages of the present invention maybe ascertained by reviewing the present disclosure and the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is further described in the detailed descriptionwhich follows, in reference to the noted plurality of drawings by way ofnon-limiting examples of exemplary embodiments of the present invention,in which like reference numerals represent similar parts throughout theseveral views of the drawings, and wherein:

FIG. 1 shows yaw steering geometry;

FIG. 2 shows orbit parameter (modified from J. R. Wertz);

FIG. 3 shows sun parameter;

FIG. 4 shows spacecraft motion for constant yaw steering option (β≧0);

FIG. 5 shows SADM motion for constant yaw steering option (β≧0);

FIG. 6 shows spacecraft motion for constant yaw steering option (β≦0);

FIG. 7 shows a SADM motion for constant yaw steering option (β≦0);

FIG. 8 shows a smoothing function f=f(η) for dynamic yaw steering;

FIG. 9 shows a dynamic yaw steering for d=10: {dot over(ψ)}_(max,d)<{dot over (ψ)}_(max) and {umlaut over (ψ)}_(max,d)<{umlautover (ψ)}_(max);

FIG. 10 shows dynamic yaw steering for d=500: {dot over(ψ)}_(max,d)>{dot over (ψ)}_(max) and {umlaut over (ψ)}_(max,d)>{umlautover (ψ)}_(max); and

FIG. 11 shows dynamic yaw steering for d=258: {dot over(ψ)}_(max,d)≦{dot over (ψ)}_(max) and {umlaut over (ψ)}_(max,d)≦{umlautover (ψ)}_(max).

DETAILED DESCRIPTION OF THE PRESENT INVENTION

The particulars shown herein are by way of example and for purposes ofillustrative discussion of the embodiments of the present invention onlyand are presented in the cause of providing what is believed to be themost useful and readily understood description of the principles andconceptual aspects of the present invention. In this regard, no attemptis made to show structural details of the present invention in moredetail than is necessary for the fundamental understanding of thepresent invention, the description taken with the drawings makingapparent to those skilled in the art how the several forms of thepresent invention may be embodied in practice.

This example presents a special Yaw Steering Guidance Law for aspacecraft (spacecraft) as applicable for the Galileo System TestbedGSTB-V2, where: spacecraft continuous nadir pointing is performed withone selected axis (z-axis); spacecraft rotation is performed around thenadir pointed axis in order to orient the spacecraft solar array normalin an optimal way towards the Sun, based on a state-of-the-art one-axissolar array drive mechanism; One selected spacecraft panel (+x panel)perpendicular to the nadir line and to the solar array axis is orientedsuch, that Sun incidence is avoided (with the exception of slidingincidence); and The two spacecraft panels (y-panels), the panel normalof which is parallel to the solar array axes, are illuminated from theSun with an incidence angle less than a predefined critical angle.

Introduction to Yaw Steering

In order to optimize the electric power generation from the solararrays, the solar array active plane has to be oriented perpendicular tothe sun line. Otherwise, the available power would approximately bereduced with the cosine of the Sun incidence angle. Based on an Earthpointed platform, the solar array pointing requires two independentrotations in general. In order to avoid rather complex two-axes SolarArray Drive Mechanisms (SADM), two well-known types of operation areestablished, depending on the Sun elevation with respect to the orbitplane:

-   -   Zero (or at least near zero) inclination, applicable for all GEO        spacecraft, where the Sun elevation with respect to the orbit        plane is ≦23.44°: only one rotation is performed with a maximum        power loss of about 8.3%. The spacecraft is operated at a        constant (0°, 180°) yaw angle (“constant yaw”) with the solar        array axis oriented parallel to the orbit normal. Only the solar        panel is rotated around its longitudinal axis using an one-axis        SADM; and    -   non-zero inclination, applicable for most LEO and MEO        satellites, where the Sun elevation with respect to the orbit        plane changes in a wide range |β|≦23.44°+i (i=inclination): the        first rotation is performed by the spacecraft itself, where the        satellite is rotated around its Earth pointing yaw-axis (z-axis,        “yaw steering”) orienting the spacecraft x/z-plane parallel to        the Sun line (i.e. Solar array axis and spacecraft y-axis        perpendicular to the Sun line). For the second rotation again an        off-the-shelf one-axes SADM is used, rotating the solar panel        around its longitudinal axis.

Yaw Steering—Kinematic Yaw Steering

The Yaw Steering law is derived from pure geometric relations asindicated in FIG. 1.

According to FIG. 1 the following “kinematic” yaw steering law can bederived (Note: arctan(x,y) in the notation here (similar to MAPLE)corresponds to the well-known function atan 2(x,y) as used in highercomputer languages, e.g. FORTRAN, Matlab, . . . ):

$\begin{matrix}{{\psi = {\arctan\left( {{\tan\;\beta},{\sin\;\eta}} \right)}}{{\sigma = {\arctan\left( {{- \sqrt{1 - {\cos^{2}{\beta \cdot \cos^{2}}\eta}}},{{- \cos}\;{\beta \cdot \cos}\;\eta}} \right)}},}} & \left( {{{eq}.\mspace{14mu} 3.1}\text{-}1} \right)\end{matrix}$where (according to the above figure):

-   -   η=η(t) describes the orbit in-plane motion (η=0 is obtained from        the projection of the Sun line to the orbit plane);    -   β=β(t) is the sun elevation with respect to the orbit plane;    -   ψ(t) is the spacecraft reference yaw angle with respect to the        LVLH coordinate system (z axis pointing towards the Earth        center); and    -   σ(t) is the solar array drive mechanism (SADM) rotation angle        (for σ=0 the solar panel normal points parallel to the        spacecraft z-axis).

The transformation to the usual Orbit and Sun parameter is given in FIG.2 (e.g. in an inertial frame centered in the Earth, X pointing towardsVernal Equinox Y, Z pointing to the North and Y augmenting to aright-hand system) where:

-   -   Ω—Right ascension of ascending node (RAAN);    -   i—Orbit inclination angle;    -   u—Argument of latitude;    -   ε=23.44°—Obliquity of ecliptic; and    -   λ—Sun seasonal angle (λ angle determined in the ecliptic plane,        λ=0° at Vernal Equinox, see FIG. 3).

The relevant yaw steering angles β and η can be calculated from:β=arcsin(sin ε·sin λ·cos i+cos λ−sin Ω·sin i−cos ε·sin λ·cos Ω·sin i)u ₀=arctan((cos Ω·cos i·cos ε·sin λ+sin i·sin ε˜sin λ−sin Ω·cos i·cosλ), (cos Ω·cos λ+sin Ω·cos ε·sin λ))η=u−u ₀  (eq. 3.1-2)

Neglecting seasonal variation and orbit plane motion during a certaintime period, obtains approximately for a circular orbit with orbitalrate ω₀:{dot over (η)}(t)≈ω₀=const.β(t)≈const.  (eq. 3.1-3)

The key element concerning yaw steering dynamics is the elevation angleβ. For GSTB-V2 and GalileoSat (orbit inclination i=56.0°) the totalvariation is 0°≦|β|≦79.44°. If the kinematic yaw-steering law would beapplied for the complete range of β, a singularity would occur for β=0°and η=0°, 180°, requiring indefinite spacecraft and SADM rotationalrates and rotational accelerations, or at least high rates andaccelerations for β≈0°. Thus, special measures have to be applied in apredefined band |β|≦β₀, assuming this band centered about β=0°.

Yaw Steering Options For Small Sun Elevation—Constant Yaw

If only power constraints are considered, for β=0° the solar arrayplanes can be oriented perpendicular to the Sun by only one (SADM)rotation, if the satellite y-axis is pointed (constantly) perpendicularto the orbit plane (like a “standard” spacecraft, e.g. GEO zeroinclination applications). If the elevation angle β is in the predefinedband ±β₀, this “standard” attitude will be maintained (“Constant Yaw”,ψ=0°), however on the cost of a slight power loss as indicated above forthe GEO applications proportional to 1-cos(β). Typical values areβ₀=10°, resulting in a tolerable power degradation of about 1.5% andmoderate spacecraft agility and SADM motion requirements.

Based on thermal requirements (explained in more detail below), for theGSTB-V2 and Galileo the critical Sun elevation angle is reduced toβ₀=2°. In the following figures, selected situations for maximum,critical ±ε (ε—“arbitrary small value”) and minimum magnitude Sunelevation angle are presented for the GalileoSat and GSTB-V2 orbit,separated for β≧0 as well as for β≦0. However, these figures are onlysketched here as an example. At the critical positive elevation, whereβ=β₀ we obtain ψ_(max)=178° and σ_(max)=0.007°/s . For the “criticalorbit angle” η=0°, 180°, 360°, . . . , we get {dot over(ψ)}_(max)=0.2°/s, {umlaut over (ψ)}_(max)=7.8 μrad/s. It is obvious,that the yaw steering profiles for β≧0 and β≦0 provide different, butsymmetric shapes with respect to the ψ=0° line, here referred to as the“positive yaw steering shape” and the “negative yaw steering shape.” Thetransition between both types can easily be performed e.g. in a certainarea near η=90°, starting from constant yaw.

With respect to FIGS. 4–7: FIG. 4 shows the spacecraft motion forConstant Yaw Steering Option (β≧0); FIG. 5 shows the SADM motion forConstant Yaw Steering Option (β≧0); FIG. 6 shows the spacecraft motionfor Constant Yaw Steering Option (β≦0); and FIG. 7 shows the SADM motionfor Constant Yaw Steering Option (β≦0).

Modified Kinematic Yaw Steering

Due to thermal constraints for GSTB-V2 and for GalileoSat two majoradditional requirements have to be considered, which significantlyinfluence the yaw steering scenario: no Sun incidence on the spacecraft+x panel shall occur (with the exception of slipping incidence); and Sunincidence on the spacecraft ±y panel shall be less than 2°.

From the first requirement it can be concluded, that yaw steering,however modified and however taking into account the system actuationcapabilities, has to be performed at any time, independent of the Sunelevation. The second requirement puts constraints on the critical angleβ₀, where the transition from Kinematic Yaw Steering law to the modifiedyaw steering law in the band |β|≦β₀ should take place.

Next, detailed information on how the two above requirements will bediscussed together with the spacecraft actuation capabilities can becombined by a dynamically shaped yaw steering law, shortly referencedhere as “Dynamic Yaw Steering.”

Dynamic Yaw Steering—Dynamic Yaw Steering Law

The basic idea concerning yaw steering in the critical band |β|≦β₀ is,to limit the spacecraft angular motion requirements at orbit angles η=0,180°, 360°, . . . (see the above-noted figures). For β=0°, in an idealcase, a Δψ=180° spacecraft flip within an infinitesimal small timeinstant would be required, meaning that indefinite spacecraft rate andacceleration would occur. In reality, of course, only very limitedspacecraft rate and angular acceleration are feasible due to theactuation system limited capabilities, i.e. torque and angular momentumlimits of the reaction wheels.

Several options could be applied, such as a simple “bang-bang” typemaximum acceleration (with predefined torque limits) and with limitedmaximum angular rate. However, excitation of solar array oscillation aswell as fuel sloshing motion should be avoided. Furthermore, forgyro-less Normal Mode operation, smooth type actuation is preferable,too.

Among a lot of dynamically smooth yaw steering laws, the following hasbasically been selected for the critical band |β|≦β₀. However, a specialprocedure has to be applied approaching β=0 (according to a change insign for β), where a smooth transient between the positive and negativeyaw steering shape has to be applied (switching logic for δ see thefurther chapter for details).

$\begin{matrix}{{{{{- \beta_{0}} \leq \beta \leq {\beta_{0}:\psi}} = {\arctan\left( {{\tan\;\beta_{d}},{\sin\;\eta}} \right)}}\beta_{d} = {\beta + {f \cdot \left( {{\beta_{0} \cdot \delta} - \beta} \right)}}},{\delta = {{{\pm 1}f} = \frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}}},{d - \text{design~~parameter}}} & \left( {{{eq}.\mspace{14mu} 3.3}{.1}\text{-}1} \right)\end{matrix}$

It should be mentioned, that the SADM steering law remains unchangedcompared to the kinematic law presented in eq. 3.1-1.

The smoothing function f=f(η) is sketched in FIG. 8 including typicalvalues for d as parameter:

The basic idea of the Dynamic Yaw Steering profile can be explained withFIG. 8 together with eq. 3.3.1-1: For β=β₀ the transient betweenKinematic and Dynamic Yaw Steering solution is obtained in a continuousway; For |β|<β₀ and in a certain range around the critical orbit angleη=0°, 180°, 360°, . . . β_(d)→β₀ is obtained. Thus, the yaw steeringmotion is smoothed to the behavior like for the critical Sun elevationangle β₀, which is the “lower” limit of the Kinematic Yaw Steeringprofile, being per definition in-line with the spacecraft actuationsystem capabilities; and for |β|<β₀ and in the dynamically non-criticalrest period along the orbit angle, the yaw steering motion is performedas for β_(d)→β, which is close to the standard kinematic solution.

Such a solution provides the advantage of a smooth yaw steering motion,avoiding any discontinuities for the yaw steering angle as well as forrate and angular acceleration, and thus also for the actuation torques.

Design of Dynamic Yaw Steering Law Parameter

One major design parameter has to be optimized for the given missionorbit, which is the parameter d in eq. 3.3.1-1. The basic idea is, notto exceed the actuation requirements from Kinematic Yaw Steering at itslimits for β=β₀. Instead of an analytic way a rather pragmatic solutionis presented here for determination of the optimum parameter:

-   -   Determination of dimensioning (but feasible) Kinematic Yaw        Steering rotational motion requirements in terms of maximum        angular rate {dot over (ψ)}_(max) and angular acceleration        {umlaut over (ψ)}_(max) by numerical evaluation of {dot over        (ψ)}32 {dot over (ψ)}(η) and {umlaut over (ψ)}={umlaut over        (ψ)}(η) for β=β₀);    -   Determination of dimensioning Dynamic Yaw Steering rotational        motion requirements in terms of maximum angular rate {dot over        (ψ)}_(max,d) and angular acceleration {umlaut over (ψ)}_(max,d)        by numerical evaluation of {dot over (ψ)}={dot over (ψ)}(η) and        {umlaut over (ψ)}={umlaut over (ψ)}(η) for β=0 and given        parameter d; and    -   Perform proper variation of d and select the optimum parameter        such, that {dot over (ψ)}_(max,d)≦{dot over (ψ)}_(max) as well        as {umlaut over (ψ)}_(max,d)≦{umlaut over (ψ)}_(max), where at        least one constraint should be fulfilled as equality.

FIGS. 9, 10 and 11 indicate the procedure, where the above results aretaken for the dimensioning requirements. The parameter d has beenselected as d=10, d=500, and after some iterations, the optimum d=258for the given orbit was obtained: FIG. 9 shows the Dynamic Yaw Steeringfor d=10: {dot over (ψ)}_(max,d)<{dot over (ψ)}_(max) and {umlaut over(ψ)}_(max,d)<{umlaut over (ψ)}_(max); FIG. 10 shows the Dynamic YawSteering for d=500: {dot over (ψ)}_(max,d)>{dot over (ψ)}_(max)postponed to the next orbit; and If |β₁|<|β₂|, the actual shape isinstantaneously switched to the complementary one, i.e. δ=sign(β₂).

However it should be noted that When β₁ or β₂=0, which is less likely, anumerically small value ≠0 should be selected.

Evaluation of Dynamic Yaw Steering Rate and Angular Acceleration

The knowledge of yaw steering angular rate and angular acceleration isrequired for: Engineering purposes; and usage as feed-forward commandsin the on-board control loops for dynamic tracking control improvement.

To avoid numerical differentiation, in particular for calculation ofin-orbit feed-forward commands, the following provides the analyticequations for angular rate and acceleration. The differentiationprocedure is rather elementary and is performed here with somesubstitutions in a consecutive way based on eq. 3.3.1-1 and eq. 3.1-3,introducing some basic mathematical conversions in order to avoidnumerical undefined expressions. For completeness the basic equationsare recalled, together with the solar array drive steering algorithms:|β|≧β₀ Kinematic Yaw Steering ƒ={dot over (ƒ)}={umlaut over(ƒ)}=0  (eq.3.3.4-0)

$\begin{matrix}{{f = \frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}}{\overset{.}{f} = {{- \left( {1 + d} \right)} \cdot \omega_{0} \cdot \frac{\sin\; 2\;\eta}{\left( {1 + {{d \cdot \sin^{2}}\eta}} \right)^{2}}}}\overset{¨}{f} = {{- 2} \cdot \left( {1 + d} \right) \cdot \omega_{0}^{2} \cdot \mspace{50mu}\frac{{\cos\; 2\;\eta} - {{d \cdot \sin^{2}}{\eta \cdot \left( {1 + {{2 \cdot \cos^{2}}\eta}} \right)}}}{\left( {1 + {{d \cdot \sin^{2}}\eta}} \right)^{3}}}} & \left( {{{eq}.\mspace{14mu} 3.3}{.4}\text{-}1} \right)\end{matrix}$and {umlaut over (ψ)}_(max,d)>{umlaut over (ψ)}_(max); FIG. 11 shows theDynamic Yaw Steering for d=258: {dot over (ψ)}_(max,d)≦{dot over(ψ)}_(max) and {umlaut over (ψ)}_(max,d)≦{umlaut over (ψ)}_(max).

Sun Elevation Zero Transient

From the above results, it can easily be concluded, that the optimumorbit angle η_(t) for transient from the positive (β>0, 0≦ψ≦+180°) tothe negative yaw steering shape (β<0, −180°≦ψ≦0) (and vice versa) wouldbe for β≅0 and either η_(t1)=90° or η_(t2)=270° (in principle). In orderto keep the yaw steering angle numerically always in the band−180°≦ψ≦+180° (avoiding complicated on-board modulo 360° operations)η_(t1) is preferred.

Thus any transient between positive and negative yaw steering shapeswill simply be performed for η_(t)=90°. However, in reality β=0 andη_(t)=90° will not occur simultaneously. To overcome this problem, thefollowing simple procedure could be applied: When approaching theη_(t)=90° point, the actual Sun elevation β₁=β(η_(t)) is calculated;subsequently the future Sun elevation β₂=β(η_(t)+360°) for oneadditional orbit is calculated from orbital elements and Sun motionpropagation; if no change in sign between both values β₁ and β₂ willoccur (β₀ ^(•)β₂>0), the actual yaw steering shape is maintained basedon the actual Sun elevation angle β, i.e. δ=sign(β₁); and if a change insign between both values β₁, β₂ occurs (β₁·β₂<0), the followingprocedure is applied: If |β₁|≧|β₂|, the actual yaw steering shape ismaintained, i.e. δ=sign(β₁) and the switching to the complementary shapeisβ_(d)=β+ƒ·(β₀·δ−β){dot over (β)}_(d)={dot over (ƒ)}·(β₀·δ−β){umlaut over (β)}_(d)={umlaut over (ƒ)}·(β₀·δ−β)δ=±1  (eq.3.3.4-2)Sun elevation zero transient together with accordingly selected valuefor δ as described above

$\begin{matrix}{{{\psi = {\arctan\left( {{\tan\;\beta_{d}},{\sin\;\eta}} \right)}}{u = {{\sin\;{\eta \cdot {\overset{.}{\beta}}_{d}}} - {{0.5 \cdot \sin}\; 2\;{\beta_{d} \cdot \cos}\;{\eta \cdot \omega_{0}}}}}{v = {1 - {\cos^{2}{\beta_{d} \cdot \cos^{2}}\eta}}}{\overset{.}{\psi} = \frac{u}{v}}\overset{.}{u} = {{\cos\;{\eta \cdot \omega_{0} \cdot {\overset{.}{\beta}}_{d}}} + {\sin\;{\eta \cdot {\overset{¨}{\beta}}_{d}}} - \mspace{45mu}{\cos\; 2\;{\beta_{d} \cdot \cos}\;{\eta \cdot \omega_{0} \cdot {\overset{.}{\beta}}_{d}}} + {{0.5 \cdot \sin}\; 2\;{\beta_{d} \cdot \sin}\;{\eta \cdot \omega_{0}^{2}}}}}{\overset{.}{v} = {{\sin\; 2\;{\beta_{d} \cdot \cos^{2}}{\eta \cdot {\overset{.}{\beta}}_{d}}} + {\cos^{2}{\beta_{d} \cdot \sin}\; 2\;{\eta \cdot \omega_{0}}}}}{\overset{¨}{\psi} = \frac{{v \cdot \overset{.}{u}} - {u \cdot \overset{.}{v}}}{v^{2}}}} & \left( {{{eq}.\mspace{14mu} 3.3}{.4}\text{-}3} \right)\end{matrix}$

In order to avoid discontinuities in the yaw angle ψ in the case whereδ·β<0, the yaw angle has to be modified according toψ_(i):=ψ_(i)−sign(ψ_(i)−ψ_(i-1))·2π if |104 _(i)−ψ_(i-1)|>π, where idenotes the instantaneously determined yaw angle, and i−1 the yaw angleone sampling period before.

The SADM angle and rate are determined from the following expressions:

$\begin{matrix}{{\sigma = {\arctan\left( {{- \sqrt{1 - {\cos^{2}{\beta \cdot \cos^{2}}\eta}}},{{- \cos}\;{\beta \cdot \cos}\;\eta}} \right)}}{\overset{.}{\sigma} = \frac{{\omega_{0} \cdot \cos}\;{\beta \cdot \sin}\;\eta}{\sqrt{1 - {\cos^{2}{\beta \cdot \cos^{2}}\eta}}}}} & \left( {{{eq}.\mspace{14mu} 3.3}{.4}\text{-}4} \right)\end{matrix}$If 1−cos² β·cos² η approaches zero, the limit value for {dot over (σ)}is given by:{dot over (σ)}=ω₀·sign(sin η)

Thus, together with the Sun elevation zero transient, as describedprevious, the total set of equations is visible for implementation.

The example describes a simple yaw steering law as a supplement to theknown kinematic yaw steering for application to Earth pointed satellites(here a cubic central body with perpendicularly assembled central bodyouter panels is assumed) combining the following features: thespacecraft is continuously Nadir pointed with one selected axis;moreover the spacecraft performs a rotation around the nadir pointedaxis in order to orientate the spacecraft solar array normal in anoptimal way towards the Sun, based on a state-of-the-art one-axis solararray drive mechanism; one selected spacecraft panel perpendicular tothe nadir line and to the solar array axis is oriented such, that anySun incidence is avoided (with the exception of sliding incidence); thetwo spacecraft panels, the panel normal of which is parallel to thesolar array axes, are illuminated from the Sun with an incidence angleless than a predefined critical angle; in the critical area of small Sunelevation angles with respect to the orbit plane (yaw angle singularity,yaw flip); the yaw steering motion is continued with application of adynamic smoothing of the rotation (Dynamic Yaw Steering), not exceedingthe spacecraft actuation system capabilities and avoiding anydiscontinuities in yaw angle, angular rate and angular acceleration;furthermore a procedure for transition from the positive to the negativeyaw steering law is derived avoiding any discontinuities in yaw angle;improvement of the closed loop attitude control dynamics trackingcapabilities, together with the yaw steering reference angle profile thereference angular rate and angular acceleration profile can easily bederived; the solar array drive mechanism steering laws remain unchangedwith respect to the known kinematic yaw steering reference profile; andthe Sun incidence requirements with respect to the spacecraft panels canbe met with a yaw error of 2° (arising e.g. from attitude control).

It is noted that the foregoing examples have been provided merely forthe purpose of explanation and are in no way to be construed as limitingof the present invention. While the present invention has been describedwith reference to an exemplary embodiment, it is understood that thewords which have been used herein are words of description andillustration, rather than words of limitation. Changes may be made,within the purview of the appended claims, as presently stated and asamended, without departing from the scope and spirit of the presentinvention in its aspects. Although the present invention has beendescribed herein with reference to particular means, materials andembodiments, the present invention is not intended to be limited to theparticulars disclosed herein; rather, the present invention extends toall functionally equivalent structures, methods and uses, such as arewithin the scope of the appended claims.

1. A method for yaw steering a spacecraft comprising: performing yawsteering of the spacecraft to have a yaw angle (ψ) for all sun elevationangles (β); and smoothing a yaw steering motion for orbital positionparameters (η) where high rotational rates {dot over (ψ)} and highrotational accelerations {umlaut over (ψ)} would occur.
 2. The methodaccording to claim 1 wherein high rotational rates {dot over (ψ)} arerotational rates exceeding a predetermined rotational rate and highrotational accelerations {umlaut over (ψ)} are rotational accelerationsexceeding a predetermined rotational acceleration.
 3. The methodaccording to claim 1, wherein the yaw steering motion is smoothed fororbit angles η=k·180° (where k=0, 1, 2, . . . ).
 4. The method accordingto claim 1, wherein the smoothing function (f) comprises a designparameter (d) which is configured to optimize a smoothing effect of thesmoothing function (f).
 5. The method according to claim 1, wherein asmooth transient between a positive yaw steering shape for β>0 and anegative yaw steering shape for β<0 is performed at a sun elevationangle β≅0 and at an orbit angle η=90°.
 6. The method according to claim1, wherein the yaw steering is performed such that a first panel of asatellite body structure is always directed away from the sun.
 7. Themethod according to claim 6, wherein a plane defined by the first panelis oriented parallel to a yaw axis of the satellite.
 8. The methodaccording to claim 1, wherein the yaw steering is performed such that asecond and a third panel of a satellite body structure always have anincident sun angle smaller than a predetermined angle.
 9. A method foryaw steering a spacecraft comprising: steering the spacecraft to have ayaw angle (ψ) for all sun elevation angles (β) in accordance with a yawsteering guidance law that provides a yaw steering motion about a yawaxis which is smooth for all sun elevation angles (β), wherein at leastpart of the guidance law comprises a smoothing function (f), which is afunction of an orbital position parameter (η) of the spacecraft, thesmoothing function (f) smoothing the yaw steering motion for values ofthe orbital position parameter (η) where high rotational rates {dot over(ψ)} or high rotational accelerations {umlaut over (ψ)} would occurwithout the smoothing function (f).
 10. The method according to claim 9wherein high rotational rates {dot over (ψ)} are rotational ratesexceeding a predetermined rotational rate and high rotationalaccelerations {umlaut over (ψ)} are rotational accelerations exceeding apredetermined rotational acceleration.
 11. A method for yaw steering aspacecraft comprising: steering the spacecraft to have a yaw angle (ψ)for all sun elevation angles (β) in accordance with a yaw steeringguidance law that provides a yaw steering motion about a yaw axis whichis smooth for all sun elevation angles (β), angle (β₀), in which thefirst part of the guidance law is independent of the wherein at leastpart of the guidance law comprises a smoothing function (f), which is afunction of an orbital position parameter (η) of the spacecraft, thesmoothing function (f) smoothing the yaw steering motion for values ofthe orbital position parameter (η) where high rotational rates {dot over(ψ)} or high rotational accelerations {umlaut over (ψ)} would occurwithout the smoothing function (f), and wherein a first part of theguidance law is applied for all sun elevation angles (β) larger or equalto a defined sun elevation angle (β₀), in which the first part of theguidance law is independent of the smoothing function (f), and a secondpart of the guidance law is applied for all sun elevation angles (β)smaller than a defined sun elevation angle (β₀), in which the secondpart of the guidance law is based on the smoothing function (f).
 12. Themethod according to claim 11, wherein a first part of the guidance lawcomprises a standard yaw steering law ψ=atan 2(tan β,sin η), which is afunction of the sun elevation angle (β) and the orbital positionparameter (η), and a second part of the guidance law comprises amodified yaw steering law ψ=atan 2(tan β_(d),sin η) which is a functionof the orbital position parameter (η) and a sun elevation angleparameter (β_(d)) and is a function of the smoothing function (f) andthe sun elevation angle (β).
 13. A method for yaw steering a spacecraftcomprising: performing yaw steering of the spacecraft to have a yawangle (ψ) for all sun elevation angles (β); smoothing a yaw steeringmotion for orbital position parameters (η) where high rotational rates{dot over (ψ)} and high rotational accelerations {umlaut over (ψ)} wouldoccur, and wherein the smoothing function (f) is defined as:$f = {\frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}.}$ such that drepresents a design parameter.
 14. The method according to claim 13,wherein the guidance law is based on the following functions: for|β|≧β₀: such that (β₀) represents a defined sun elevation angle;ƒ=0  (1) and for|β|<β₀: $\begin{matrix}{f = \frac{\cos^{2}\eta}{1 + {{d \cdot \sin^{2}}\eta}}} & (2)\end{matrix}$β_(d)=β+ƒ·(β₀·δ−β), δ=±1  (3)ψ=atan 2(tan β_(d),sin η).  (4) such that (β_(d)) represents a sunelevation angle parameter.
 15. A method for yaw steering a spacecraftcomprising: performing yaw steering of the spacecraft to have a yawangle for all sun elevation angles; and reducing rotational rates orrotational accelerations below a predetermined value by smoothing a yawsteering motion at predetermined orbital position parameters.
 16. Themethod according to claim 15 further comprising: steering the spacecraftwith a yaw steering guidance law which provides a yaw steering motionabout a yaw axis, wherein at least part of the guidance law comprises asmoothing function that is a function of an orbital position parameterof the spacecraft, the smoothing function smoothing the yaw steeringmotion.
 17. A method for yaw steering a spacecraft comprising:performing yaw steering of the spacecraft to have a yaw angle for allsun elevation angles; reducing rotational rates or rotationalaccelerations below a predetermined value by smoothing a yaw steeringmotion at predetermined orbital position parameters; steering thespacecraft with a yaw steering guidance law which provides a yawsteering motion about a yaw axis; wherein at least part of the guidancelaw comprises a smoothing function that is a function of an orbitalposition parameter of the spacecraft, the smoothing function smoothingthe yaw steering motion, and wherein a first part of the guidance law isapplied for all sun elevation angles larger or equal to a defined sunelevation angle, in which the first part of the guidance law isindependent of the smoothing function, and a second part of the guidancelaw is applied for all sun elevation angles smaller than a defined sunelevation angle, in which the second part of the guidance law is basedon the smoothing function.